We now consider how to solve the coupled Bloch equations (Eqs. REF  and ). At first, we simplify them. Unlike the near-zero field case where noble gas spins and electron spins are strongly coupled together and operate as a comagnetometer [1]}, we consider a different situation where \(^{87}\) Rb and \(^{129}\) Xe spins are under a large bias field \(B_{0}^{z}\)  along \(z\)  (above 200 nT).
In this situation, \(^{87}\) Rb and \(^{129}\) Xe spins are weakly coupled together, greatly simplifying Eqs. REF  and . The details are explained as follows. (1) \(P^{n}_{z}\)  can be approximated as a constant, which is much larger than \(P^{n}_{x}\)  and \(P^{n}_{y}\) . Based on Eq. , the \(^{129}\) Xe spins evolve under the static field of the effective field \(\lambda M^{e} {P}_z^{e}\)  along \(z\)  generated by \(^{87}\) Rb spins and the bias field \(B^{z}_{0}\) . Because the applied \(B_{0}^{z}\)  is much larger than the effective field \(\lambda M^{e} {P}_z^{e}\) , we can first neglect the term \(\lambda M^{e} {P}_z^{e}\)  in Eq.  for simplicity (but the \(^{87}\) Rb effective field along \(z\)  should be considered when the Larmor frequency of \(^{129}\) Xe is calibrated, see Sec. REF ). As a result, Eq.  can be independently solved at first and \(\textbf {P}^n\)  can be evaluated. (2) Then, we can take the solution of \(\textbf {P}^n\)  into Eq. REF  and solve for \(\textbf {P}^e\) .
